Multishell model for Majumdar-Papapetrou spacetimes
Metin Gu¨rses*Mathematics Department, Bilkent University, 06800 Ankara-Turkey Burak Himmetog¯lu†
Physics Department, Bilkent University, 06800 Ankara-Turkey
(Received 4 May 2005; published 29 July 2005; publisher error corrected 2 August 2005) Exact solutions to static and nonstatic Einstein-Maxwell equations in the presence of extremely charged dust embedded on thin shells are constructed. Singularities of multi-black-hole Majumdar- Papapetrou and Kastor-Traschen solutions are removed by placing the matter on thin shells. Double spherical thin shell solution is given as an illustration and the matter densities on the shells are derived.
DOI:10.1103/PhysRevD.72.024032 PACS numbers: 04.40.Nr, 04.20.Jb
I. INTRODUCTION
It was shown that the Majumdar-Papapetrou (MP) solu-tion [1,2] describes a spacetime possessing N extremely charged black holes [3]. Later, extremely charged static dust sources for MP spacetimes were considered [4,5] and it was shown that solutions to Einstein-Maxwell equations with extremely charged dust restricted to thin shells exist. Moreover, solutions for various geometries including pla-nar, spherical and cylindrical shells were obtained [5]. It was also shown that spherical shells can be used as sources for the Extreme Reissner-Nordstro¨m (ERN) spacetimes [5]. The cosmological black-hole solutions which are time-dependent generalizations of the MP solution were also studied [6,7] and generalizations with discussions on nonradiative character of these spacetimes were discussed in [8].
In linear theories mass density of point particle distribu-tion can be expressed as r Pimir ai where mi’s
are the masses of each particle, r is the Dirac delta function and ai’s are the locations of the point particles.
This expression is in consistence with the Poisson’s equa-tion, r2V 4 with V X N i1 mi jr aij: (1) Such a point particle representation of the mass density is not valid in nonlinear theories. For instance, in a nonlinear theory like r2V 4V3 0, there exists no solution V
for Pimir ai. We overcome the above problem by replacing massive particles by spherical thin shells with centers at r aiand radii ri. Then the mass density can now be represented as r PN
i10iFi where
Fir 0 i 1; 2; . . . ; N represents the positions of the
shells and 0iare the mass density of each thin shell. The
space is divided into N 1 spaces. In our proposed
method we shall solve such potential problems where the solutions are continuous everywhere.
In this work, we will first review the solutions to Einstein-Maxwell equations in the presence of extremely charged static dusts restricted to thin shells that are given in [5]. Then we will generalize the methods in [5] to find solutions possessing extremely charged dusts restricted to multiple shells and obtain singularity free versions of MP-solutions given in [3]. We will also work out the double shell problem explicitly and show that the matter distribu-tions on the shells are not uniform and that interior of the shells are not flat unlike the single shells in [5]. Finally we will work on cosmological MP solutions discussed in [6 – 8] and develop similar multishell models.
II. SHELL MODELS FOR THE MAJUMDAR-PAPAPETRAU SPACETIMES
LetM be a four dimensional spacetime with the metric,
ds2 2dt2 2h
ijdxidxj; (2)
where hij is an Euclidean 3-metric and is a function of spatial coordinates xi only. With this metric assumption,
we consider a charged static dust source with the matter part of the energy-momentum tensor given by,
TM uu; (3)
where u 1
0
since the dust is static. The
four-potential due to the static charged dust is in the form A
A0xi0, so the current four-vector is given by,
J xiu xi0
; (4)
where xi is the charge density of the dust. Therefore the
electromagnetic and the Maxwell energy-momentum ten-sors are given by,
F rA rA; (5) M 1 4 F F 1 4F F g : (6)
*Electronic address: gurses@fen.bilkent.edu.tr †Electronic address: himmet@ug.bilkent.edu.tr
Using the Einstein tensor calculated from the metric (2) and the field equations G 8TM
M with (3)–
(6) we get,
A0!
; ! 1: (7)
Then plugging the above equation back into the Einstein equations give,
r2 43 0: (8)
Therefore the Einstein-Maxwell equations reduce to a non-linear type of Poisson’s equation. To find the equation satisfied by the charge density xi we use the Maxwell equations rF 4Jwith (4) and (5) to get,
r2 4!3 0: (9)
Comparing (8) and (9) gives ! which shows that the dust source is extremely charged.
One can solve the reduced Einstein-Maxwell Eqs. (8) and (9) for the extremely charged dust restricted to a thin shell as given in [5]. Letting S be a regular surface in R3
defined by S x; y; z 2 R3; Fx; y; z 0 , the matter
density can be written as,
r 0rFr; (10)
where 0 is the matter distribution on the shell S.
Generalizing the method given in [5] we can set,
r 0r 1r F 2r1 F; (11)
where F is the Heaviside step function. The above choice can be made since the spacetimeM is now divided into two disjoint spacetimesMandMby S where the metric functions are 0 1 and 0 2 respectively. We also require that the following equations are satisfied,
r2
0 r21 r22 0; (12)
1 2jS 0: (13) These follow from the fact that in M and M the Laplace’s equation is satisfied as can be seen from (8) (Since 0 in M and M) and the metric must be continuous across the shell. Inserting (11) into (8) and using the conditions (12) and (13) we get,
0
r1 r2 rF
43 0
jS: (14)
Let S be the sphere F r a 0. One can choose 2 0, 1 3%; &1
a 1
r which clearly satisfies (12) and
(13). Using (14) gives,
0%; & 3%; & 4a2
0jS3
; (15)
where 3%; & satisfies,
1 r2sin% @ @% sin% @ @% 1 r2sin2% @2 @&2 3%; & 0: At this point we would like to remark that the above Eq. (15) can also be obtained by using the Israel junction conditions [9]. However, when Israel junction conditions are used the surface energy-momentum tensor has the components,
Stt
3%; &
4a20jS4; S%% S&& 0: (16)
So the matter density Ion S is given by using the fact that
Stt IututI 2 0 jS, I%; & 3%; & 4a2 0jS2 : (17)
As can be seen from (17) in Israel method we get 2 0in the
denominator instead of 3
0 as in (15) which is due to a
dimensional scaling. The total mass on the shell can be written as an integral of 0Fover the 3 spatial
dimen-sions which is equal to the integral of I over the 2
dimensional surface, that is, Z V3 0F3d3x Z S2 I2d2x; (18)
So we get the same total mass on the shell in either method which means that there is no ambiguity.
When the interior of the sphere S is chosen as a flat spacetime, we can set 0 1 and 3 m0 as constant
which turns out to be the mass of the shell. So (15) becomes,
0 m0
4a2: (19)
So the interior and exterior metric functions are given by,
out 1 m0 a m0 r ; (20) in 1; (21)
where m0 is the mass of the shell as can clearly be seen from (19). This solution represents the ERN solution ex-terior to a spherical shell with flat inex-terior. Letting r R m0= where 1 m0=awith m0 aassumed,
the exterior metric can be written as,
ds2 21 m0 R 2 dt2 1 m0 R 2 dR2 R2d2; (22) which is the conventional form of ERN metric (after a scaling). As we noted earlier, we aim to remove the singu-larity of an ERN spacetime. Choosing m0< awe see from
(20) that out> 0 is always satisfied. Comparing (22), the
relation r R m0= and (20) one can realize that the singularity of ERN spacetime is at out 0 (which
sponds to R 0) and the horizon is at r 0 (which corresponds to R m0). This means that by restricting the matter on the shell with m0< awe remove the both the singularity and the horizon of the ERN spacetime. In the case m0> a(20), out 0 is not excluded. The case
m0 a represents the Levi-Civita-Bertotti-Robinson
(LCBR) spacetime outside the shell and flat spacetime inside. By letting r m2
0=Rthe exterior metric is obtained
as,
ds2 m0
R2 dt
2 dR2 R2d2 ; (23)
as the usual conformally flat LCBR metric. At this point we would like to note that in our analysis the limiting case a ! 0 does not exist since in that limit the matter density (10) would become r 0rr and this choice is incon-sistent with the Einstein-Maxwell Eqs. (8) and (9). If we had the Newtonian theory, such a limit mathematically would be consistent with the Poisson’s equation. The non-existence of the a ! 0 limit is in agreement with the results of [10].
To remove the singularities of a MP spacetime possess-ing N ERN black holes discussed in [3], we place the matter source on thin shells. A MP spacetime possessing
NERN black holes has the metric function given by,
1 X
N
j1
mi
jr aij; (24) where aiis the position of the ithERN black hole. Such a
spacetime contains N number of singularities. We can place the extremely charged dust on N spatially separated shells that do not intersect, instead of considering point sources. Generalizing the choice we made in (10) we can write the matter density of the spacetime as,
r X
N
j1
0jrFj; (25)
where 0jis the matter distribution on the jthshell defined
by Sj:Fjr 0. We can also generalize the choice for
as, 0 e YN j1 Fj XN j1 j 1 Fj ; (26)
so that the metric function inside the jthshell is given by,
inj 0 j 8j 1; . . . ; N: (27) Moreover the metric function exterior to all of the N shells is given by,
ext
0 e: (28)
The functions 0, jand e satisfy the following,
r2
0 r2j r2e 0; (29)
e jjSj 0 8j 1; . . . ; N; (30) where we again use the fact that the Laplace’s equation is satisfied in source free regions and the metric across the shells must be continuous. Using (26)–(30) we calculate r2as,
r2 X N
j1
rj re rFjFj: (31)
Using the fact that Fj Fk Fj for k j, 3 is
given by,
3 X N
j1
0j0 j3Fj: (32)
Then inserting (31) and (32) into (8) we get,
0jre rj rFj 40 j3
jSj: (33) As an illustration we consider the case of two spherical shells with radii r1and r2and the centers located at a1and a2. Our aim is to obtain the exterior MP solution with the metric function given in (24) so that the singularities can be removed. The equations defining the spherical shells are given by F1 jr a1j r1 0 and F2 jr a2j
r2 0. Making analogy with (24) and the single shell
solution (20) we choose, e m1 r1 m2 r2 m1 jr a1j m2 jr a2j: (34) Appropriate choices for 1 and 2 can be made by the following forms using (29) and (30),
1 m2 r2 m2 jr a2j; (35) 2 m1 r1 m1 jr a1j; (36)
which in turn gives the full metric functions with the choice
0 1 as, ext 1 m1 r1 m2 r2 m1 jr a1j m2 jr a2j; (37) in1 1 m2 r2 m2 jr a2j; (38) in 2 1 m1 r1 m1 jr a1j: (39) The above choices describe an exterior MP spacetime and ERN spacetimes inside each shell. Let a1 aez and a2 aez so that the centers of the shells are on the z-axis with r1 r2< 2a. The surface matter distributions
01 m1 4r2 1 1 m2 r2 m2 r2 1 4ar1cos%1 4a2 q 3 ; (40) 02 m2 4r2 2 1 m1 r1 m1 r2 2 4ar2cos%2 4a2 q 3 ; (41)
where %1 and %2 are spherical coordinates on S1 and S2 with their centers are taken as if origin. It can clearly be seen from (40) and (41) that the existence of a second shell disturbs the uniform matter distribution on the other shell so that the system stays at equilibrium. The above equa-tions can be obtained by Israel method as we discussed before, where one obtains the second power of the term in the denominator instead of the third power as in (40) and (41) which is due to dimensional scaling. One can obtain (24) by letting 1 m1 r1 m2 r2 in (37) so ext becomes, ext 1 m1 jr a1j m2 jr a2j ext; (42) where m1 m1= and m2 m2= . The MP spacetime possessing two ERN black holes has two horizons each described as a surface enclosing the points where the black holes are located [3]. These singularities are defined by the vanishing of ext. So choosing the masses and the radii of
the shells such that m1=r1 m2=r2< 1, we guarantee that
ext> 0 from (37), which means that by this choice we
remove the singularities. Note that in
1 in (38) and in2 in
(39) can never vanish in their domains of definition. The case m1=r1 m2=r2> 1 is irrelevant since we can not
remove the singularities of the MP spacetime. The case
m1=r1 m2=r2 1 should correspond to two mass
gen-eralization of the LCBR metric. Thus we obtained the double ERN black-hole solutions with singularities re-moved. (Note that we obtained the solution with a rescaling factor ). At this point we remark that MP-solutions with multiple spherical dust shells as sources, can not have flat interiors as we showed in (38) and (39). Moreover, the shells disturb each other and cause angular dependence of matter density on the shells. One can see from (40) and (41) that the second shell (S2) has maximum matter density at its north pole and the first shell (S1) has its maximum matter density at its south pole. Such a configuration puts the system in equilibrium. One can also calculate the masses of the shells by integrating (40) and (41) on S1
and S2respectively as in (18) which gives the mass of the
first shell as m1 and the second shell as m2 which is
expected. We again note that the limits r1 ! 0 and r2 !
0 do not exist since the matter density (25) is not consistent with the Einstein-Maxwell Eqs. (8) and (9). This means that we can not have a source density of the form,
r m1r a1 m2r a2: This is again in total agreement with [10].
One can also consider other solutions to the Laplace’s equation for the metric function to obtain various space-times. It was shown that such spacetimes possess naked singularities [3]. One can remove these naked singularities by placing the extremely charged source on thin shells as we did in this work for multiple ERN black-hole solutions. The same procedure can also be applied to the stationary generalization of MP-spacetimes given by Israel-Wilson [11] and Perje´s [12] to remove the naked singularities of these spacetimes.
III. SHELL MODELS FOR THE KASTOR-TRASCHEN SPACETIMES
It was shown that time-dependent generalizations of MP spacetimes exist and solutions corresponding to N ex-tremely charged comoving black holes in a de-Sitter back-ground were considered in [6 –8]. The idea of removing the singularities of N comoving black-hole solutions by thin shells were considered briefly in [7] for testing the cosmic censorship conjecture. At this point we would like to extend the discussion given in [7] by the methods we discussed above.
The cosmological MP solution which is a time-dependent generalization of the MP solution with metric (2) in cosmological coordinates is given by,
ds2 ~U2dt2 Rt2U~2h
ijdxidxj; (43)
where Rt is the scale factor and hij is an Euclidean
metric. For a single extreme Reissner-No¨rdstrom-de-Sitter (ERNdS) black hole with its charge equal to its mass Q M the metric function ~U in the metric (43) is given by,
~
U 1 eHtM
r : (44)
We can obtain the ERNdS solution in static coordinates by following the transformation given in [6] Given the metric (43) with ~U being arbitrary we consider a comoving charged dust as source. As before, we assume the four-potential of the form A A00
and u U1~0 since
the dust is comoving (i.e. static in cosmological coordi-nates). Then the Einstein equations with positive cosmo-logical constant become,
G 8uu M g; (45)
where Mis the Maxwell tensor given as in (6). Then the Einstein equations (45) give,
_R R 2 3; (46) r2U 4Rt~ 2U~3 0; (47)
~
U 1 1
Rtx; (48)
where dot over R represents derivative with respect to t and
xis independent of t. The above equations are obtained by separating the cosmological and electromagnetic-matter parts of the Einstein equations. One can note that (46) is just the Friedmann equation for a flat cosmology. Solving (46) yields Rt eHt where H p 3=. As
discussed in [6 –8] negative H corresponds to black-hole spacetimes while positive H corresponds to white hole spacetimes. We consider negative H for the rest of the paper (H jHj). The Maxwell equations rF
4J give,
r2U ~ 4
! Rt
2U~3 0; (49)
with ! 1 and being the charge density of the comov-ing dust. Comparcomov-ing (47) and (49) we get ! as before, so we conclude that the dust is extremely charged. Letting d1 Rtdt we can write the metric (43) as,
ds2 U2d12 U2h
ijdxidxj: (50)
where
U H1 x; (51)
r2U 4U3 0: (52)
One can see from the Einstein Eqs. (52) that the product
U3must be time independent. This is clear from the fact r2Uis time independent as can be seen from (51) and to
have the Eq. (52) satisfied for all times we must have U3
to be time independent.
The above derivation of Einstein-Maxwell equations for a charged dust was given in [7]. Our treatment of the shell model will be parallel to which we considered in the previous part of this work, that is different from the shells considered in [7]. We start with a dust model and then take its limit to thin shells with the use of Dirac delta functions for the matter density. Since (47) and (49) are totally same with Eqs. (8) and (9) we can consider similar multishell models with the matter density given in (25) and the metric function U as in (26) (Clearly we just replace with U). One difference is that the metric function U is time depen-dent via H1. However, as can be seen from (51) and (52) this time dependence does not affect Einstein-Maxwell equations. In source free regions we again have the Laplace’s equation satisfied. The equation of each surface defining the shells are again given by Fjr 0. Thus we follow the lines (25) to (33) to get the same dependence for the surface matter density.
As an example, we again consider two spherical shells with radii r1, r2 and with centers located at a1, a2.The equations describing the shells are given by,
F1 jr a1j r1 0; F2 jr a2j r2 0:
(53) Because of linearity of Laplace’s equation, we extend the single ERNdS solution in cosmological coordinates given in (50) to the double ERNdS case for the exterior metric function as, Uext H1 m1 r1 m2 r2 m1 jr a1j m2 jr a2j; (54) where we have chosen U0 H1 in (28). Then (38) and (39) suggests, Uin 1 H1 m2 r2 m2 jr a2j ; (55) Uin 2 H1 m1 r1 m1 jr a1j; (56) for the metrics inside the first and the second shells. Clearly the choices (54)–(56) satisfy the required conditions (29) and (30) for N 2. We have mentioned above that U3
must be time independent which is satisfied in our multi-shell model since this product is given with the use of (25), (26), (55), and (56) as,
U3 01Uin
13F1 02Uin23F2:
Since F1;2 are time independent and 01;2Uin
1;2 will be
shown to be time independent below.
Again choosing a1 aez and a2 aez such that
r1 r2< 2a the surface matter distributions are
calcu-lated from (33) as,
01 m1 4r2 1 H1 m2 r2 m2 r2 1 4ar1cos%1 4a2 q 3 ; 02 m2 4r2 2 H1 m1 r1 m1 r2 2 4ar2cos%2 4a2 q 3 ;
where the definitions of %1 and %2 are given as before. As
can be seen from above, 01 and 02 are time dependent but when they are integrated on the surface of each shell (the term 01;2Uin
1;23. will appear in the integral which is
time independent), they give constant masses for the shells as m1 and m2 which is in consistence with the result of Bonnor [8] . If we had a single shell, the interior of the shell would be de-Sitter. But for the case of double shells (and for N > 2 also) as can be seen from (55) and (56) the interior of the shells are ERNdS. Unless this choice is made, the continuity condition of the metric across the shells will be violated. This fact was not realized in [7] where the interior of the shells were considered as de-Sitter.
To remove the singularities of the cosmological multi-black hole solutions we again restrict the matter on thin shells. For that purpose, we assume that the thin shells are formed at 1 0 with H < 0 so from (54) one can see that
Uext never vanishes for later times 1 0. This means that
the exterior metric is singularity free for all times. Moreover the interior of the shells described by (55) and (56) can also never vanish for the choice we made. Thus the whole spacetime becomes regular. The spacetime distances jjPQjjgbetween two neighboring (space) points P and Q is given by jjPQjjg jUjjjP Qjj where jjP Qjj is the distance in R3, and U is given in (51). Hence in exterior
and interior spacetimes the space distances increase with constant (Hubble) speed. For this reason the shells inflate without collision, because the distances among them also increase.
There is also another interesting possibility of construct-ing regular spacetimes by considerconstruct-ing H > 0 for 1 0 and
H < 0 for 1 0. With these choices one can see that Uext,
Uin
1 and Uin2 never vanishes and the spacetime metric is
continuous at 1 0. In this model, the Hubble constant H changes sign in each universe separated by 1 0 which causes a delta-function type of singularity at this hypersur-face. Spacetimes (interior and exterior spacetime regions) are contracting for 1 0 and expanding for 1 0.
Our method can be applied also to higher dimensional generalizations of the Majumdar-Papapetrau-Kastor-Traschen solutions [13] to study brane world creations and collisions [14].
IV. CONCLUSION
We considered thin mass shell models for the Majumdar-Papapetrou and Kastor-Traschen spacetimes. In both cases we found exact solutions of the field equa-tions describing gravitational fields of the extremely charged N-number mass shells. Shell interiors are also curved spacetimes matching smoothly to the exteriors through (three dimensional) infinitely thin shells. We found the mass ( also the charge) densities on each shell. The solutions obtained this way are free of singularities. For the case of Kastor-Traschen spacetimes choosing the sign of the Hubble constant H properly we showed that shells are moving away from each other with constant velocity.
Shell models in higher dimensional theories seem to be also very interesting. In order that our method to be appli-cable to such theories one has to modify Einstein field equations with the inclusion of a dust matter as we did in this work where the mass is distributed on thin shells. In this case the shells are also higher dimensional. In the case of five dimensions for instance the thin shells are four dimensional spacetimes. Our work on this matter will be communicated elsewhere.
ACKNOWLEDGMENTS
This work is partially supported by the Scientific and Technical Research Council of Turkey and by the Turkish Academy of Sciences.
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